Quantum and classical localization, the spin quantum Hall effect, and generalizations

We consider network models for localization problems belonging to symmetry class C. This symmetry class arises in a description of the dynamics of quasiparticles for disordered spin-singlet superconductors which have a Bogoliubov-de Gennes Hamiltonian that is invariant under spin rotations but not under time reversal. Our models include but also generalize the one studied previously in the context of the spin quantum Hall effect. For these systems we express the disorder-averaged conductance and density of states in terms of sums over certain classical random walks, which are self-avoiding and have attractive interactions. A transition between localized and extended phases of the quantum system maps in this way to a similar transition for the classical walks. In the case of the spin quantum Hall effect, the classical walks are the hulls of percolation clusters, and our approach provides an alternative derivation of a mapping first established by Gruzberg, Ludwig, and Read.